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Properties

Label 349.3.d.a
Level $349$
Weight $3$
Character orbit 349.d
Analytic conductor $9.510$
Analytic rank $0$
Dimension $116$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [349,3,Mod(136,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.136");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 349.d (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.50956122617\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 116 q + 32 q^{6} - 18 q^{7} - 30 q^{8} - 352 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q + 32 q^{6} - 18 q^{7} - 30 q^{8} - 352 q^{9} - 16 q^{10} + 8 q^{11} + 32 q^{12} + 22 q^{13} + 56 q^{14} + 32 q^{15} - 436 q^{16} - 112 q^{18} - 52 q^{19} - 72 q^{20} + 50 q^{21} + 200 q^{23} + 144 q^{24} - 408 q^{25} + 72 q^{26} - 84 q^{28} - 6 q^{30} + 68 q^{31} - 130 q^{32} + 146 q^{33} + 62 q^{34} - 60 q^{35} + 2 q^{38} + 64 q^{39} - 218 q^{40} + 144 q^{41} - 180 q^{43} + 32 q^{44} + 122 q^{46} - 100 q^{47} + 56 q^{50} - 172 q^{51} + 296 q^{52} - 188 q^{53} - 540 q^{54} + 160 q^{55} - 140 q^{58} + 202 q^{59} + 62 q^{61} + 424 q^{62} + 362 q^{63} + 206 q^{65} - 268 q^{66} + 252 q^{67} - 404 q^{68} + 12 q^{71} + 664 q^{72} - 68 q^{74} - 884 q^{77} + 50 q^{79} + 1284 q^{81} - 666 q^{82} - 742 q^{84} + 1240 q^{85} + 220 q^{87} + 60 q^{88} + 48 q^{89} + 298 q^{90} + 940 q^{94} - 1588 q^{96} + 180 q^{97} + 704 q^{98} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −2.71975 + 2.71975i − 1.11998i − 10.7940i 6.04670i 3.04607 + 3.04607i 1.40472 + 1.40472i 18.4780 + 18.4780i 7.74563 −16.4455 − 16.4455i
136.2 −2.62430 + 2.62430i 3.62850i − 9.77395i − 2.36293i −9.52228 − 9.52228i 3.10965 + 3.10965i 15.1526 + 15.1526i −4.16599 6.20105 + 6.20105i
136.3 −2.56902 + 2.56902i − 5.86126i − 9.19973i 1.81287i 15.0577 + 15.0577i −2.87189 − 2.87189i 13.3582 + 13.3582i −25.3544 −4.65730 − 4.65730i
136.4 −2.55729 + 2.55729i 0.472866i − 9.07943i − 4.97290i −1.20925 − 1.20925i −2.07087 − 2.07087i 12.9896 + 12.9896i 8.77640 12.7171 + 12.7171i
136.5 −2.49330 + 2.49330i − 3.23492i − 8.43313i − 5.67501i 8.06564 + 8.06564i 7.68343 + 7.68343i 11.0531 + 11.0531i −1.46472 14.1495 + 14.1495i
136.6 −2.39413 + 2.39413i 2.04228i − 7.46367i 0.636178i −4.88948 − 4.88948i −9.46071 − 9.46071i 8.29247 + 8.29247i 4.82907 −1.52309 − 1.52309i
136.7 −2.26165 + 2.26165i − 3.24564i − 6.23016i − 7.79815i 7.34051 + 7.34051i −8.13843 − 8.13843i 5.04385 + 5.04385i −1.53416 17.6367 + 17.6367i
136.8 −2.21621 + 2.21621i 0.657841i − 5.82318i 8.54748i −1.45792 − 1.45792i 4.05882 + 4.05882i 4.04056 + 4.04056i 8.56724 −18.9430 − 18.9430i
136.9 −2.19097 + 2.19097i 4.58132i − 5.60070i 7.33796i −10.0375 − 10.0375i −7.41766 − 7.41766i 3.50708 + 3.50708i −11.9885 −16.0773 − 16.0773i
136.10 −2.14234 + 2.14234i − 3.12273i − 5.17928i 4.61720i 6.68997 + 6.68997i −0.704114 − 0.704114i 2.52643 + 2.52643i −0.751454 −9.89163 − 9.89163i
136.11 −1.96815 + 1.96815i 5.58966i − 3.74724i − 6.45976i −11.0013 − 11.0013i −2.12610 − 2.12610i −0.497467 − 0.497467i −22.2443 12.7138 + 12.7138i
136.12 −1.82701 + 1.82701i 1.94275i − 2.67593i − 3.27601i −3.54943 − 3.54943i 6.46085 + 6.46085i −2.41908 − 2.41908i 5.22572 5.98530 + 5.98530i
136.13 −1.81814 + 1.81814i 4.05826i − 2.61127i 3.87315i −7.37849 − 7.37849i 5.69408 + 5.69408i −2.52491 − 2.52491i −7.46951 −7.04192 − 7.04192i
136.14 −1.80494 + 1.80494i − 3.61385i − 2.51563i 2.18027i 6.52279 + 6.52279i 0.294777 + 0.294777i −2.67921 − 2.67921i −4.05993 −3.93526 − 3.93526i
136.15 −1.64211 + 1.64211i − 0.308921i − 1.39308i 1.35572i 0.507284 + 0.507284i −4.00593 − 4.00593i −4.28086 − 4.28086i 8.90457 −2.22625 − 2.22625i
136.16 −1.47413 + 1.47413i 0.0665614i − 0.346137i − 9.57939i −0.0981204 − 0.0981204i 1.39338 + 1.39338i −5.38628 − 5.38628i 8.99557 14.1213 + 14.1213i
136.17 −1.40891 + 1.40891i − 3.28998i 0.0299657i − 3.24762i 4.63528 + 4.63528i 5.25090 + 5.25090i −5.67784 − 5.67784i −1.82398 4.57560 + 4.57560i
136.18 −1.16521 + 1.16521i 2.72727i 1.28457i − 6.62972i −3.17784 − 3.17784i −3.22846 − 3.22846i −6.15764 − 6.15764i 1.56202 7.72501 + 7.72501i
136.19 −1.05551 + 1.05551i − 4.90084i 1.77180i 8.82260i 5.17288 + 5.17288i −6.08908 − 6.08908i −6.09219 − 6.09219i −15.0182 −9.31234 − 9.31234i
136.20 −0.919924 + 0.919924i − 5.73484i 2.30748i − 4.88883i 5.27562 + 5.27562i 3.06475 + 3.06475i −5.80240 − 5.80240i −23.8884 4.49735 + 4.49735i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.3.d.a ✓ 116
349.d odd 4 1 inner 349.3.d.a ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.3.d.a ✓ 116 1.a even 1 1 trivial
349.3.d.a ✓ 116 349.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(349, [\chi])\).